On the preserved extremal structure of Lipschitz-free spaces

Abstract

We characterize preserved extreme points of the unit ball of Lipschitz-free spaces $\mathcal {F}({X})$ in terms of simple geometric conditions on the underlying metric space $(X,d)$. Namely, the preserved extreme points are the elementary molecules corresponding to pairs of points $p,q$ in $X$ such that the triangle inequality $d(p,q)\leq d(p,r)+d(q,r)$ is uniformly strict for $r$ away from $p,q$. For compact $X$, this condition reduces to the triangle inequality being strict. As a consequence, we give an affirmative answer to a conjecture of N. Weaver that compact spaces are concave if and only if they have no triple of metrically aligned points, and we show that all extreme points are preserved for several classes of compact metric spaces $X$, including Hölder and countable compacta.